Chapter 3 - The Concept of Differentiation

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1 alculus hapter - The oncept o Dierentiation Applications o Dierentiation opyright preptests4u.com. All Rights Reserved. This Academic Review is brought to you ree o charge by preptests4u.com. Any sale or trade o this review is strictly prohibited.

2 alculus h : The oncept o Dierentiation Applications o Dierentiation Applications o Dierentiation Applications o dierentiation to the ollowing topics are given throughout the eamples. a. ritical Points b. Local Maimum and Minimum c. Absolute Maimum and Minimum d. Intervals o Increasing and Decreasing e. Inlection Points. Test or oncavity g. The Mean Value Theorem Assumptions: For a-g, it is assumed that the unction ( ) is continuous on a closed interval[ a,b]. For g, in addition to continuity, it is also assumed that unction ( ) is dierentiable on the open interval ( a,b). a. ritical points: To ind the critical point(s) o ( ), let ( ) 0 or ( ) ±. b. Absolute Maimum and Minimum / Local Maimum and Minimum: To ind absolute Maimum or Minimum, evaluate ( ) at a,b, and all critical points. The highest value is absolute maimum and the lowest value is absolute minimum. The local maimum or/and minimum are determined by evaluating at the critical points. ( ) ( ) I sign o changes rom positive to negative on a given point ( a, b) ( ) c, then has a local maimum at c. ( ) ( a, b) ( ) I sign o changes rom negative to positive on a given point c, then has a local minimum at c. c. Intervals o Increasing and Decreasing: I ( ) 0 on a given interval, then ( ) I ( ) p 0 on a given interval, then ( ) is increasing in that interval. is decreasing in that interval. Page o opyright preptests4u.com. All Rights Reserved. This Academic Review is brought to you ree o charge by preptests4u.com. Any sale or trade o this review is strictly prohibited.

3 alculus h : The oncept o Dierentiation Applications o Dierentiation d. Test or oncavity: ( ) 0 ( ) I or all belong to a given interval, then the graph o is concave up or upward on this interval. ( ) p 0 ( ) I or all belong to a given interval, then the graph o is concave down or downward on this interval. e. Second derivative tests or local minimum and local maimum: I c is a critical point, ( c) 0, and ( ) 0, then ( ) c at. I c is a critical point, ( c) 0, and ( ) p 0, then ( ) c at.. The Mean Value Theorem: I ( ) is continuous on [ b] There is a point c ( a, b ) () c a, and ( ) is dierentiable on ( b) such that: ( b) ( a) b a has a local maimum has a local minimum a,, then Eample : Find the local maimum and minimum or the unction ( ) ( ) ( + ) ( ) 0,, critical points Second derivative test or local minimum and maimum,. ( ) 0 ( ) 4 0, hence local minimum at 4 p 0, hence local maimum at Page o opyright preptests4u.com. All Rights Reserved. This Academic Review is brought to you ree o charge by preptests4u.com. Any sale or trade o this review is strictly prohibited.

4 alculus h : The oncept o Dierentiation Applications o Dierentiation Eample : Find the absolute maimum and minimum or the unction ( ) interval[ 0,]. ritical points ound in eample,, Since [ 0, ], it is discarded. ( 0), Absolute Maimum ( ), Absolute Minimum () 4 given in eample on the Eample : Find the value(s) o at which unction ( ) e has relative etrema, (local Ma./Min.). ( ) e e e ( ) 0 ±, critical points, e 0 Second derivative test or local minimum and maimum, ( ) 4e + ( ) e ( ) ( ) e p 0, Local Maimum at 0, Local Minimum at Page 4 o opyright preptests4u.com. All Rights Reserved. This Academic Review is brought to you ree o charge by preptests4u.com. Any sale or trade o this review is strictly prohibited.

5 alculus h : The oncept o Dierentiation Applications o Dierentiation Eample 4: Find the absolute maimum and minimum or the unction ( ), in eample, on the interval [,0]. Note: ( ) [, 0] Function is continuous on. Discard. And, Absolute Minimum e ( 0 ) 0, e ( ), Absolute Maimum Eample 5: Find the maimum and minimum or the unction ( ) cos sin on the interval [ 0,π ]. Notice that ( ) is continuous on [ 0,π ]. ( ) cos sin cos ( cos )( sin ) 0 π π cos 0, critical points. sin 0 sin p, hence no solution. Page 5 o opyright preptests4u.com. All Rights Reserved. This Academic Review is brought to you ree o charge by preptests4u.com. Any sale or trade o this review is strictly prohibited.

6 alculus h : The oncept o Dierentiation Applications o Dierentiation Remember: sin ( 0 ), Absolute Maimum π, Absolute Minimum ( π ), Absolute Maimum Eample 6: (The Mean Value Theorem) Apply the Mean Value Theorem to the ollowing unction on the given interval. + ( ), [,5] First check i the conditions or using The Mean Value Theorem apply: Note:.05 (, 5) ( ) [,5] ( ) (,5) a. is continuous on b. is dierentiable on Both conditions are satisied, hence () and ( ) +. 5 ( ) ( 5) ( ) , Page 6 o opyright preptests4u.com. All Rights Reserved. This Academic Review is brought to you ree o charge by preptests4u.com. Any sale or trade o this review is strictly prohibited.

7 alculus h : The oncept o Dierentiation Applications o Dierentiation Eample 7: (The Mean Value Theorem) heck i The Mean Value Theorem is applicable to the ollowing unction on the given interval. + ( ), [,] First check i the conditions or using The Mean Value Theorem apply: ( ) is not continuous at 0 [,], hence The Mean Value Theorem does not apply. Note: I a unction is continuous at a point, then it is not necessarily dierentiable at that point. But i a unction is dierentiable at a point, then the unction is certainly continuous at that point. See the net eample. Eample 8: heck i The Mean Value Theorem is applicable to the ollowing unction on the given interval. e +, 0 ( ), + + 4, p 0 [, + ] First check i the conditions or using The Mean Value Theorem apply: ( ) [, + ] ( ) dierentiable at 0 (, + ) a. Function is continuous on b. Function is not. onclusion: The Mean Value Theorem does not apply here due to part b. See Eample 9, Section, hapter or the details o the dierentiability. Page 7 o opyright preptests4u.com. All Rights Reserved. This Academic Review is brought to you ree o charge by preptests4u.com. Any sale or trade o this review is strictly prohibited.

8 alculus h : The oncept o Dierentiation Applications o Dierentiation Eample 9: Find the intervals, at which unction ( ) 8 and concave down. is increasing, decreasing, concave up ( ) 6 4 ( 8) ( 4)( + ) 0,4, ritical Points ( ) 6 6 6( ) 0,, Inlection Point Interval o Decreasing, ( ) p 0, (,4) Interval o Increasing, ( ) 0, (, ) U ( 4, + ) Interval o oncave up, ( ) 0, (, ) Interval o oncave down, ( ) p 0, (,) Note: From the given intervals, one may easily answer the ollowing questions: Eercise: ( ) ( 4, ) ( ) (,) a. Interval at which is increasing and concave up: b. Interval at which is decreasing and concave down: ( ) ( ) c. Interval at which is increasing and concave down: d. Interval at which is decreasing and concave up: Eample 0: A rectangular piece o land has dimensions by y. We want to build a symmetric rectangular a y tennis court inside the land which has a margin rom and marginb rom on both sides. Assuming the area o the land is given as a. onstraint y square eet, ind: Page 8 o opyright preptests4u.com. All Rights Reserved. This Academic Review is brought to you ree o charge by preptests4u.com. Any sale or trade o this review is strictly prohibited.

9 alculus h : The oncept o Dierentiation Applications o Dierentiation b. Maimum area o the tennis court ( a) ( y b) A t, also y a A t 4 ( a) b b + ab a a A t b + 0 b + a 0, y b b a Eample : y h In a right triangle with hypotenuse, adjacent side and ied height, ind the rate o change o y with respect to the rate o change o. y + () t y y( t) h, and Using chain rule: dy d d y + 0 d dy y d ost Function. ost unction is denoted by ( ). c. Average cost unction ( ). Marginal cost unction, c ( ) ( ) 4. I average cost is minimum, then Marginal cost Minimum o average cost q, where is number o units produced. 5. Demand unction, price unction, ( ) 6. Revenue unction R( ) q( ) 7. Marginal Revenue unction R ( ) Page 9 o opyright preptests4u.com. All Rights Reserved. This Academic Review is brought to you ree o charge by preptests4u.com. Any sale or trade o this review is strictly prohibited.

10 alculus h : The oncept o Dierentiation Applications o Dierentiation 8. Proit unction P( ) R( ) ( ) 9. Marginal proit unction P ( ) R ( ) ( ) 0. Maimize proit P ( ) 0 ( ) R ( ). Supply unction, P ( ) sup ply [ P P ( ) ] d. Producer surplus at equilibrium point: where P is the price at the 0 equilibrium point, calculated by inding the equilibrium point and substituting this into the demand unction. To ind the equilibrium point, let demand unction equals the supply unction. s Eample : The demand unction or a product is given as q( ) ( ) Find: 00 and the cost unction is 000 a. Average ost Function c ( ) ( ) b. Marginal ost Function ( ) c. Revenue Function R ( ) q( ) Page 0 o opyright preptests4u.com. All Rights Reserved. This Academic Review is brought to you ree o charge by preptests4u.com. Any sale or trade o this review is strictly prohibited.

11 alculus h : The oncept o Dierentiation Applications o Dierentiation d. Proit Function P ( ) R( ) ( ) e. Number o Units which maimizes the proit ( ) R ( ) ( ) R ( ) From this Find the rate o change o revenue in terms o rate o change o the units produced at a given production level dr R ( ) d n d dr ( n) R ( n) d n d 0 n Page o opyright preptests4u.com. All Rights Reserved. This Academic Review is brought to you ree o charge by preptests4u.com. Any sale or trade o this review is strictly prohibited.